Linear Rescaling to Accurately Interpret Logarithms

The standard approximation of a natural logarithm in statistical analysis interprets a linear change of $p$ in $\ln(X)$ as a $(1+p)$ proportional change in $X$, which is only accurate for small values of $p$. I suggest base-$(1+p)$ logarithms, where $p$ is chosen ahead of time. A one-unit change in $\log_{1+p}(X)$ is exactly equivalent to a $(1+p)$ proportional change in $X$. This avoids an approximation applied too broadly, makes exact interpretation easier and less error-prone, improves approximation quality when approximations are used, makes the change of interest a one-log-unit change like other regression variables, and reduces error from the use of $\log(1+X)$.